The Nonsplit Resolving Domination Polynomial of a Graph

Abstract

Metric representation of a vertex v in a graph G with an ordered subset = { 1 , 2 , … , } of vertices of G is the kvector ( | ) = ( ( , 1 ), ( , 2 ), … , ( , )), where ( , ) is the distance between and in G. The set is called a Resolving set of , if any two distinct vertices of have distinct representation with respect to . The cardinality of a minimum resolving in s called a dimension of , and is denoted by ( ). In a graph = ( , ), A subset ⊆ s a nonsplit resolving dominating set of if it is a resolving, and nonsplit dominating set of . The minimum cardinality of a nonsplit resolving dominating set of is known as a nonsplit resolving domination number of , and is represented by ( ). In network reliability domination polynomial has found its application [20], a resolving set has diverse applications which includes verification of network and its discovery, mastermind game, robot navigation, problems of pattern recognition, image processing, optimization and combinatorial search [19]. Here, we are introducing nonsplit resolving domination polynomial of . Some properties of the nonsplit Resolving domination polynomial of are studied and nonsplit resolving domination polynomials of some well-known families of graphs are calculated.